Definite Integral and Trapezoidal Estimate $--$ Solve Me: Definite Integral and Trapezoid: Hand Calculation

In the graph to the right, the red curve is a plot of the function curve y $=$ f$($x$),$ where f$($x$)=$cos$($x$)+3$ In the steps below, we use the definite integral to calculate the pink area under the curve on the interval $-1\mathrm{.}3<$ x $<0\mathrm{.}5$

To check the answer, we use the blue trapezoid to give a close estimate of the area. From the plot, we can see if the trapezoid gives an overestimate or an underestimate of the pink area.

y $=$ f$($x$)$

$3\mathrm{.}5$

$2\mathrm{.}1$

$2$

$1\mathrm{.}5$

$1$

$0\mathrm{.}5$

$0$

$-1\mathrm{.}3$

$-0\mathrm{.}4$

X

$0\mathrm{.}5$

In Steps $1$ and $2$ use the trapezoid to estimate the area under the curve y $=$ f$($x$)$ represented by the following definite integral, and tell whether the estimate is an overestimate or an underestimate

In Steps $3$ and $4$ calculate the definite integral. Finally in Step $5,$ use the trapezoidal estimate to check your integral.

In Steps $3$ and $4$ calculate the definite integral. Finally in Step $5,$ use the trapezoidal estimate to check your integral.

Step $3.$

Enter the formula for the indefinite integral. Include the constant of integration, C$.$

$($cos$($x$)+3)$ dr

Step $4.$

Use the Fundamental Theorem and the result found in Step $3$ to calculate the exact value of the pink area under the curve y $=$ f$($x$)$ represented by the definite integral. Show all work, and do not simplify or round at this step.

$0\mathrm{.}5-1\mathrm{.}3($cos$($x$)+3)$ dx

Step $5.$

Step $6.$

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Conclusion. Use a decimal approximation to compare the definite integral found in Step $4$ with the trapezoid estimate found in Step $2.$ Enter your answers, good to four digits after the decimal point. Your answer should agree with the prediction made in Step $1.$

$0\mathrm{.}5($cos$($x$)+3)$ dx $-1\mathrm{.}3$

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Trapezoid Area

Note. Calculator in Radians Mode for this Problem.